Agenda

  • Build a Multiple Linear Regression model
  • Analysis of the model
# First things first , clean your R environment
rm(list = ls(all=TRUE))
# setwd("")       # Set the working directory

Loading Libraries

library(corrplot)
## Warning: package 'corrplot' was built under R version 3.4.2
## corrplot 0.84 loaded
library(DMwR)
## Loading required package: lattice
## Loading required package: grid

Multiple Linear Regression Model

Read or load data

dataFile = "boston.csv"
boston_housing_data = read.csv(dataFile)
attach(boston_housing_data) # by attaching the dataframe to the memory, we can directly use the variable names

Data Desctiption

Data description The Boston data frame has 506 rows and 14 columns.

This data frame contains the following columns:

  • CRIM : per capita crime rate by town.

  • ZN : proportion of residential land zoned for lots over 25,000 sq.ft.

  • INDUS : proportion of non-retail business acres per town.

  • CHAS : Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).

  • NOX : nitrogen oxides concentration (parts per 10 million).

  • RM : average number of rooms per dwelling.

  • AGE : proportion of owner-occupied units built prior to 1940.

  • DIS : weighted mean of distances to five Boston employment centres.

  • RAD : index of accessibility to radial highways.

  • TAX : full-value property-tax rate per $10,000.

  • PT : pupil-teacher ratio by town.

  • B : 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town.

  • LSTAT : lower status of the population (percent).

  • MV : median value of owner-occupied homes in $1000s.

EDA - Exploratory Data Analysis and Preprocessing steps

  1. Look at the number of rows and columns
  2. Column names
  3. Look at the structure of the dataset using the function str()
  4. Look at the summary of the dataset using the summary()
  5. Checking for ‘NA’ values

1. Look at the number of rows and columns

dim(boston_housing_data)
## [1] 506  14

2. Column names

colnames(boston_housing_data)
##  [1] "CRIM"  "ZN"    "INDUS" "CHAS"  "NOX"   "RM"    "AGE"   "DIS"  
##  [9] "RAD"   "TAX"   "PT"    "B"     "LSTAT" "MV"

3. Look at the structure of the dataset using the function str()

str(boston_housing_data)
## 'data.frame':    506 obs. of  14 variables:
##  $ CRIM : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ ZN   : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ INDUS: num  2.31 7.07 7.07 2.18 2.18 ...
##  $ CHAS : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ NOX  : num  0.538 0.469 0.469 0.458 0.458 ...
##  $ RM   : num  6.57 6.42 7.18 7 7.15 ...
##  $ AGE  : num  65.2 78.9 61.1 45.8 54.2 ...
##  $ DIS  : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ RAD  : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ TAX  : int  296 242 242 222 222 222 311 311 311 311 ...
##  $ PT   : num  15.3 17.8 17.8 18.7 18.7 ...
##  $ B    : num  397 397 393 395 397 ...
##  $ LSTAT: num  4.98 9.14 4.03 2.94 5.33 ...
##  $ MV   : num  24 21.6 34.7 33.4 36.2 ...

4. Look at the summary of the dataset using the summary()

summary(boston_housing_data)
##       CRIM                ZN             INDUS            CHAS        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08205   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       NOX               RM             AGE              DIS        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       RAD              TAX              PT              B         
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      LSTAT             MV       
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00

5. Checking for ‘NA’ values

sum(is.na(boston_housing_data))
## [1] 0

Converting a varible explicity to factor

boston_housing_data$CHAS_F = factor(boston_housing_data$CHAS)
#dummies::dummy(boston_housing_data$CHAS_F)
boston_housing_data$CHAS_F1 = dummies::dummy(boston_housing_data$CHAS_F)[,1]
boston_housing_data$CHAS_F = NULL

Scatter plot with a few selected attributes

pairs(~CRIM+ZN+INDUS+NOX+RM+LSTAT,data=boston_housing_data,main="Scatterplot matrix with selected attributes")

Scatter plot with all attributes

pairs(~.,data=boston_housing_data,main="Scatterplot matrix")

Correlation

correlation_XPairwise = cor(boston_housing_data[, !(colnames(boston_housing_data) %in% c("MV","CHAS_F1"))])
corrplot(correlation_XPairwise)

Write the pairwise correlation matrix to a csv file for later analysis

write.table(correlation_XPairwise,file="pairwiseCorrelations1.csv",row.names=FALSE,col.names=FALSE,sep=",")

Model building

Simple Linear Regression

lineFitSinglePredictor = lm(MV ~ LSTAT, data = boston_housing_data); # fit a simple linear regression model
summary(lineFitSinglePredictor)
## 
## Call:
## lm(formula = MV ~ LSTAT, data = boston_housing_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -15.168  -3.990  -1.318   2.034  24.500 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 34.55384    0.56263   61.41   <2e-16 ***
## LSTAT       -0.95005    0.03873  -24.53   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.216 on 504 degrees of freedom
## Multiple R-squared:  0.5441, Adjusted R-squared:  0.5432 
## F-statistic: 601.6 on 1 and 504 DF,  p-value: < 2.2e-16

Plot the scatter plot and the fitted line for single variable

plotTitle = paste("Boston housing dataset : \n Median House Value vs %Lower Status Population\n Correlation coefficient = ",toString(cor(boston_housing_data$MV,boston_housing_data$LSTAT)),"\n MV = ",toString(lineFitSinglePredictor$coefficients[1]), "+ LSTAT*(",toString(lineFitSinglePredictor$coefficients[2]),")\n\n",sep=" ")
plot(boston_housing_data$LSTAT,boston_housing_data$MV,main=plotTitle,xlab="%Lower Status Population (LSTAT)",ylab="Median House Value (MV)")
abline(lineFitSinglePredictor,col="steelblue",lty=1,lwd=4)

Summary of the model

summary(lineFitSinglePredictor)
## 
## Call:
## lm(formula = MV ~ LSTAT, data = boston_housing_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -15.168  -3.990  -1.318   2.034  24.500 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 34.55384    0.56263   61.41   <2e-16 ***
## LSTAT       -0.95005    0.03873  -24.53   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.216 on 504 degrees of freedom
## Multiple R-squared:  0.5441, Adjusted R-squared:  0.5432 
## F-statistic: 601.6 on 1 and 504 DF,  p-value: < 2.2e-16

Residual plots for the models

plot(lineFitSinglePredictor,main="Linear model with a single predictor")

Cook’s Distance plot for simple linear regression

cd = cooks.distance(lineFitSinglePredictor)
plot(cd,main=paste("boston_housing_data"," : \n Cook's distances for each sample for the Simple Linear Model",sep=" "),xlab="Index",ylab="Cook's distance")
grid(10,10,lwd=2)

Multiple Linear Regression

Build Model

lineFitMultiplePredictors = lm(boston_housing_data$MV ~ B+CRIM+RM+LSTAT)

Summary of the model

summary(lineFitMultiplePredictors) 
## 
## Call:
## lm(formula = boston_housing_data$MV ~ B + CRIM + RM + LSTAT)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -18.016  -3.494  -1.223   1.986  29.419 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -7.941106   3.498223  -2.270 0.023629 *  
## B            0.010247   0.002968   3.452 0.000602 ***
## CRIM        -0.074057   0.032766  -2.260 0.024237 *  
## RM           5.389120   0.440138  12.244  < 2e-16 ***
## LSTAT       -0.535983   0.048740 -10.997  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 5.431 on 501 degrees of freedom
## Multiple R-squared:  0.6541, Adjusted R-squared:  0.6513 
## F-statistic: 236.8 on 4 and 501 DF,  p-value: < 2.2e-16

Residual plots for the multiple linear regression model

plot(lineFitMultiplePredictors,main="Linear model with a multiple predictor")

Regression using all predictors

lineFit_All = lm(MV ~ ., data = boston_housing_data) 
summary(lineFit_All)
## 
## Call:
## lm(formula = MV ~ ., data = boston_housing_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -15.595  -2.730  -0.518   1.777  26.199 
## 
## Coefficients: (1 not defined because of singularities)
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.646e+01  5.103e+00   7.144 3.28e-12 ***
## CRIM        -1.080e-01  3.286e-02  -3.287 0.001087 ** 
## ZN           4.642e-02  1.373e-02   3.382 0.000778 ***
## INDUS        2.056e-02  6.150e-02   0.334 0.738287    
## CHAS         2.687e+00  8.616e-01   3.118 0.001925 ** 
## NOX         -1.777e+01  3.820e+00  -4.651 4.25e-06 ***
## RM           3.810e+00  4.179e-01   9.116  < 2e-16 ***
## AGE          6.922e-04  1.321e-02   0.052 0.958229    
## DIS         -1.476e+00  1.995e-01  -7.398 6.01e-13 ***
## RAD          3.060e-01  6.635e-02   4.613 5.07e-06 ***
## TAX         -1.233e-02  3.760e-03  -3.280 0.001112 ** 
## PT          -9.527e-01  1.308e-01  -7.283 1.31e-12 ***
## B            9.312e-03  2.686e-03   3.467 0.000573 ***
## LSTAT       -5.248e-01  5.072e-02 -10.347  < 2e-16 ***
## CHAS_F1             NA         NA      NA       NA    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.745 on 492 degrees of freedom
## Multiple R-squared:  0.7406, Adjusted R-squared:  0.7338 
## F-statistic: 108.1 on 13 and 492 DF,  p-value: < 2.2e-16

Regression using interaction terms

lineFit_Interaction = lm(MV ~ LSTAT * AGE, data = boston_housing_data) 
summary(lineFit_Interaction)
## 
## Call:
## lm(formula = MV ~ LSTAT * AGE, data = boston_housing_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -15.806  -4.045  -1.333   2.085  27.552 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 36.0885364  1.4698355  24.553  < 2e-16 ***
## LSTAT       -1.3921169  0.1674555  -8.313 8.78e-16 ***
## AGE         -0.0007209  0.0198792  -0.036   0.9711    
## LSTAT:AGE    0.0041560  0.0018518   2.244   0.0252 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.149 on 502 degrees of freedom
## Multiple R-squared:  0.5557, Adjusted R-squared:  0.5531 
## F-statistic: 209.3 on 3 and 502 DF,  p-value: < 2.2e-16
plot(lineFit_Interaction)

Regression using non-linear transformation of predictors

lineFit_NonLinearTransform = lm(MV ~ CRIM + NOX + RM + DIS + PT + B + LSTAT + I(LSTAT ^ 2), data=boston_housing_data)

Regression using non-linear transformation of predictors along with a categorical predictor

lineFit_NonLinearTransform_Categorical = lm(MV ~ CRIM + CHAS_F1 + NOX + RM + DIS + PT + B + LSTAT + I(LSTAT ^ 2), data=boston_housing_data) 
summary(lineFit_NonLinearTransform_Categorical)
## 
## Call:
## lm(formula = MV ~ CRIM + CHAS_F1 + NOX + RM + DIS + PT + B + 
##     LSTAT + I(LSTAT^2), data = boston_housing_data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -17.753  -2.617  -0.333   1.750  26.172 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  38.638373   4.530364   8.529  < 2e-16 ***
## CRIM         -0.096228   0.027673  -3.477 0.000551 ***
## CHAS_F1      -2.828852   0.789583  -3.583 0.000374 ***
## NOX         -11.399562   2.973082  -3.834 0.000142 ***
## RM            3.608807   0.372924   9.677  < 2e-16 ***
## DIS          -1.175195   0.148926  -7.891 1.93e-14 ***
## PT           -0.704267   0.104296  -6.753 4.07e-11 ***
## B             0.006909   0.002446   2.824 0.004928 ** 
## LSTAT        -1.720591   0.119663 -14.379  < 2e-16 ***
## I(LSTAT^2)    0.034516   0.003207  10.761  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.394 on 496 degrees of freedom
## Multiple R-squared:  0.7758, Adjusted R-squared:  0.7717 
## F-statistic: 190.7 on 9 and 496 DF,  p-value: < 2.2e-16
plot(lineFit_NonLinearTransform,main="Linear model with multiple predictors and non-linear transformations")

Predictions on test data

set.seed(123) # for reproducible results
sample.size <- floor(0.75 * nrow(boston_housing_data))
train.index <- sample(seq_len(nrow(boston_housing_data)), size = sample.size)
train <- boston_housing_data[train.index,]
test <- boston_housing_data[- train.index,]
testPredictions_simple_linear_model = predict(lineFitSinglePredictor, test, interval = "confidence") 
testPredictions_multiple_linear_model = predict(lineFit_All, test, interval = "confidence") 
## Warning in predict.lm(lineFit_All, test, interval = "confidence"):
## prediction from a rank-deficient fit may be misleading

Evaluation of models

Simple Linear Regression

regr.eval(trues = test[,c("MV","LSTAT")],preds = testPredictions_simple_linear_model)
##        mae        mse       rmse       mape 
##  1202.9424 20145.8429   141.9361   140.7144

Mutiple Linear Regression

regr.eval(trues = test,preds = testPredictions_multiple_linear_model)
##         mae         mse        rmse        mape 
##    8105.694 2529351.586    1590.394         Inf